Work & Energy Calculator

Work and Energy Guide

What do I need to know about Energy Forms and Conservation?

Work is energy transferred by a force acting over a distance: W = F × d × cos(θ), where θ is the angle between force and displacement. The joule (J) is the SI unit: 1 J = 1 N applied over 1 m. Conservation of energy: total mechanical energy (KE + PE) remains constant in the absence of friction. A falling ball converts PE to KE — at the midpoint, half is KE and half PE. At impact (assuming no air resistance), all PE has converted to KE.

What's the key thing to understand about Kinetic Energy?

KE = ½mv². A 70 kg person running at 5 m/s (18 km/h): KE = ½ × 70 × 25 = 875 J. A 1,000 kg car at 30 m/s (108 km/h): KE = ½ × 1000 × 900 = 450,000 J = 450 kJ. Notice that doubling speed quadruples KE (because v is squared) — this is why road speed limits exist. At 60 mph, a car has 4× the KE of 30 mph, requiring 4× the stopping distance assuming equal braking force.

What should I know about Gravitational Potential Energy?

GPE = mgh, where g = 9.81 m/s² near Earth's surface. A 70 kg person climbing a 10 m staircase: GPE = 70 × 9.81 × 10 = 6,867 J ≈ 6.9 kJ ≈ 1.64 food calories. This is why stair climbing burns relatively few calories from height alone — the body is also moving forwards and working against friction. A 100 kg boulder at the top of a 100 m cliff: GPE = 100 × 9.81 × 100 = 98,100 J ≈ 98 kJ.

What's the key thing to understand about Power?

Power is the rate of doing work: P = W/t, measured in watts (W). 1 W = 1 J/s. A 75 kg person climbing a flight of stairs (3m height) in 2 seconds: W = 75 × 9.81 × 3 = 2,207 J. Power = 2207/2 = 1,104 W ≈ 1.1 kW. Professional cyclists sustain ~400 W (FTP) and peak at 1,200–1,500 W for sprint efforts. The human body is approximately 25% efficient at converting food energy to mechanical work — the rest becomes heat.

What's the key thing to understand about the Work-Energy Theorem?

The work-energy theorem states that the net work done on an object by all forces acting on it equals the change in that object's kinetic energy: W_net = ΔKE = KE_final − KE_initial. This provides an extremely useful shortcut for solving motion problems: rather than needing to know an object's acceleration and the time over which it acts, you can instead use just the work done (force times distance, for a constant force) to directly find how an object's speed has changed. For example, if a 1,000 kg car starts at rest and a net force does 125,000 joules of work on it, the work-energy theorem gives ΔKE = 125,000 J, so ½(1000)v² = 125,000, giving v² = 250, so v ≈ 15.8 m/s — found without needing to know the acceleration or the time taken at all. The theorem also works in reverse: if an object's kinetic energy decreases (it slows down), the net work done on it during that process must have been negative, meaning the net force opposed its direction of motion, exactly as happens when brakes do negative work on a moving car to bring it to a stop.

What should I know about Elastic Potential Energy?

Beyond gravitational potential energy, energy can also be stored in a stretched or compressed elastic object, such as a spring, following Hooke's Law. The elastic potential energy stored in an ideal spring is E = ½kx², where k is the spring constant (a measure of the spring's stiffness, in newtons per metre) and x is the distance the spring is stretched or compressed from its natural, unstressed length. This formula has the same mathematical form as kinetic energy (½mv²) and the energy stored in a charged capacitor (½CV²) — all three describe a quantity that builds up as the square of a displacement or quantity, a pattern that recurs throughout physics whenever a restoring force is proportional to displacement. For example, a spring with a spring constant of 200 N/m, compressed by 0.1 m, stores E = ½(200)(0.1)² = ½(200)(0.01) = 1 joule of elastic potential energy. This stored energy converts to kinetic energy as the spring releases, powering everything from a simple jack-in-the-box toy to pogo sticks, archery bows, and mechanical watch mainsprings.

What do I need to know about Work Done by a Variable Force?

The simple W = Fd formula only applies when force is constant along the direction of motion. When force varies — as it does stretching a spring, where F = kx grows with displacement — work must be found as the area under a force-versus-displacement graph, formally W = ∫F dx. For a spring, this integral works out to the familiar elastic potential energy formula ½kx², which is exactly the triangular area under its straight-line F-x graph; for more complex, non-linear forces, the same area-under-the-curve idea still applies even when there's no neat closed-form formula.

Worked Example: Work-Energy Theorem on a Braking Car

A 1,500 kg car traveling at 20 m/s brakes to a complete stop over a distance of 40 m. The work-energy theorem says the net work done on the car equals its change in kinetic energy: W = ΔKE = KE_final − KE_initial = 0 − (½ × 1,500 × 20²) = −½ × 1,500 × 400 = −300,000 J. The negative sign simply means the net force (friction from the brakes) acts opposite to the direction of motion, removing energy from the car. Since work is also force times distance, the average braking force can be found by rearranging W = F × d: F = W/d = 300,000 / 40 = 7,500 N. This is a useful double-check method for braking problems — rather than working out force and acceleration from kinematics first, the energy approach gets straight from speed and distance to the force involved. If the stopping distance were cut in half (20 m) at the same starting speed, the required force would double to 15,000 N, because the same 300,000 J of energy has to be removed over half the distance.

Table: Energy Forms and Their Formulas

Mechanical energy problems usually involve converting between a handful of energy forms. Having the formulas side by side makes it easier to spot which one applies to a given scenario.

Efficiency and Energy Losses in Real Systems

The work-energy theorem assumes all the work done goes into changing kinetic energy, but real systems always lose some energy to friction, air resistance, heat, or sound — which is why efficiency matters in practical engineering. Efficiency is defined as useful energy output divided by total energy input, usually expressed as a percentage: efficiency = (useful energy out / energy in) × 100%. A car engine that converts 100,000 J of chemical energy from fuel into only 25,000 J of useful kinetic energy at the wheels is operating at 25% efficiency — the rest is lost mostly as heat through the exhaust and engine block, a figure that's actually typical for gasoline engines. An electric motor, by contrast, might convert 90% or more of its electrical energy input into mechanical work, because it avoids the combustion losses altogether. This is why switching between energy forms always involves checking whether you're calculating the ideal (frictionless) case or the real case: a pendulum swinging back to a lower height than it started, or a ball bouncing to a lower height each time, is losing a measurable fraction of its mechanical energy to air resistance and heat at each cycle.